Pathwise Learning of Stochastic Dynamical Systems with Partial Observations

The reconstruction and inference of stochastic dynamical systems from data is a fundamental task in inverse problems and statistical learning. While surrogate modeling advances computational methods to approximate these dynamics, standard approaches typically require high-fidelity training data. In many practical settings, however, the data are indirectly observed through noisy and nonlinear measurements. The challenge lies not only in approximating the coefficients of the SDEs, but in simultaneously inferring the posterior updates given the observations. In this work, we present an amortized path generation method to address these challenges and solve nonlinear stochastic filtering from noisy observations. We first derive a variational inference formulation that solves filtering distribution for a given noisy observation path. This leads to a controlled SDE representation in which the feedback control is identified through the score structure of a pathwise Zakai equation. Motivated by this representation, we construct a conditional generative model that learns, in an amortized manner over observation paths, to transport a prior latent path measure toward the corresponding posterior path measure. We demonstrate the method on nonlinear stochastic systems with multimodal posterior structure, chaotic dynamics, and sparse observations, showing that the learned conditional path generator enables uncertainty quantification for both filtering marginals and trajectory-dependent functionals.